Chapter 2 Command Line Switches

2.1 Man page, On-line help & Usage

Endo has a wide variety of command line arguments which are processed at startup and which determine the initial conditions, dynamics, properties viewed, and so on. No command line switches are necessary as endo contains default values for all of the properties that can be set on the command line. In addition, all of these properties and many more can be interactively controlled via the keyboard and mouse once endo is running. Chapter 3 deals with interactive run-time control.

A terse summary of these command switches is provided in the UNIX-style manual page endo.LOCAL (installed in /usr/local/man ). On-line help is also available at any time by pressing the '?' or 'h' keys while endo is running. In addition, a brief usage message is displayed in the presence of an unknown argument or if "-u" is given on the command line. A more detailed explanation of the many command line switches is given here.

2.2 Selecting the map

The distributed endo software package comes preconfigured with 29 different endomorphisms of the plane. In order to select among these, you can either use the -T command line option or start endo with no options and choose a map from the menu. The syntax for specifying a map with the -T option is "-T #" where # represents a number between 0 and 28. For example, " endo -T 25" would select the MiraQuad map described below. Precise definitions of these maps are given in Appendix B. There is no default map. A map must be specified either on the command line or by selecting one from the map menu.

2.3 Specifying parameter values

Endo supports up to 24 distinct parameters. Most of the predefined maps listed in Appendix B contain only 2 or 3 parameters. You can specify parameter values on the endo command line via the switches "-ABCDEFGIJKLMabcdefgijklm value". That is, to specify a value of 2.3 for the second parameter you would pass the argument "-B 2.3". To specify the use of the MiraQuad map with parameter values 0.7 and -0.9 respectively, the command would be "endo -T 25 -A 0.7 -B -0.9". To determine the default parameter values for a map, consult Appendix C.

2.4 Selecting the dynamical view

When investigating a particular map it is sometimes desirable to fix parameter values and vary initial conditions while at other times it may be necessary to fix initial conditions and vary parameter values. It may also be helpful to display critical curves, critical arcs, precritical curves, point trajectories and other information. All of these can be specified on the command line.

In order to fix parameter values and vary initial conditions, use the "-8 0" option. To vary parameter values while fixing an initial condition, use "-Q m n" where m and n represent parameter numbers. For instance, to vary parameters 2 and 3 in the Leslie Matrix, use "-T 2 -Q 2 3". The first parameter listed is used as the x-axis and the second as the y-axis. The default view is to vary parameter values and fix an initial condition. The default parameters to vary are "-Q 0 1".

There are a number of ways to combine the -8 and -9 switches to present various aspects of the dynamics. The -8 switch can be followed by 0, 1, 2, 3, or 4 specifying the following :

-8 0 displays the basins of attraction

-8 1 bifurcation diagrams only

-8 2 bifurcation diagrams and Lyapunov sets

-8 3 same as 2 with initial condition taken as last point calculated on the attractor

-8 4 same as 2 with initial condition on critical curve-8 4

The -9 switch can be followed by a 0, 1, 2, 3, or 4 as follows :

-9 0 no Mandelbrot sets displayed

-9 1 display Lyapunov exponents in the Mandelbrot set

-9 2 display the Mandelbrot set in black

-9 3 display the rate of attraction of a critical point

-9 4 use the reserved portion of the palette for display of the Mandelbrot set

Fixing parameter values and varying initial conditions produces a graphical display of a portion of the basins of attraction. Fixing initial conditions and varying parameter values produces a graphical representation of the Lyapunov exponents which vary as the parameter values vary. In either case, a graphical representation of the point trajectories-p will also be displayed unless the "-p" option has been specified.

If in addition to the basins of attraction you wish to view the critical curves-P , use the "-P 1" switch. The -P switch acts as follows :

-P 1 display critical curves-P only

-P 2 display critical and precritical curves-P

-P 3 display precritical curves only

-P 4 display pre-images of the diagonal only

-P 5 indicates the existence of two attractors separated by the diagonal

-P 6 display pre-images of the origin only

-P 7 display pre-images of the diagonal and origin

-P 8 display pre-images of critical curves and origin

-P 9 indicates the existence of two attractors separated by the y-axis

-P 10 indicates the existence of two attractors separated by the x-axis

-P 11 use distance from origin for color indicator

-P 12 use angle with x-axis as color indicator

-P 13 indicates the existence of an attractor in each quadrant of the plane

2.5 Specifying initial conditions, settle and dwell

In order to iterate a point in the plane, it's necessary to know the coordinates of the point, the map used (including parameter values) and the number of iterates desired. Endo uses two integers, referred to as the settle and dwell , to determine the number of iterates. A point is iterated for settle + dwell iterations through the map. For the first settle iterations, no points are plotted. After allowing this "passing of transients", the subsequent dwell number of iterates are plotted. To specify the settle and dwell on the command line, use the "-S #" and "-n #" arguments to endo. The default settle and dwell is set to 256 each.

When varying the parameter values and fixing the initial conditions, to specify the coordinates of the point to iterate, use the " -X x-value -Y v-value" switches to iterate the point (x-value, y-value). An argument of -R will cause random initial conditions to be generated.

When fixing the parameter values and varying the initial conditions, use the following :

-x xval1 , xval2 , xval3

-y yval1 , yval2 , yval3

-w wval1 , wval2 , wval3

-h hval1 , hval2 , hval3

This will specify a rectangular domain of iteration with lower left corner at (xval1, yval1) and upper right corner at (xval1 + wval1, yval1 + hval1). Endo will traverse this rectangular subset of the plane, iterating each of the points in a grid. The grid size is determined by the size of the window (see -W and -H below).

The optional arguments above will be used to set the rectangular boundaries for the point trajectory-x, -y, -w, -h and critical curve-x, -y, -w, -h windows respectively. For instance, it may be useful to iterate the box (0, 1) x (1, 3) while viewing the trajectory phase space (0.25, 0.35) x (0.1, 0.4). Providing command-line arguments of " -x 0 , 0.25 -y 1 , 0.1 -w 1 , 0.1 -h 2 , 0.3" would accomplish this.

Consult Appendix D to determine the default initial conditions for a particular map.

2.6 Setting window properties

Endo contains 16 color palettes which were found to be useful in displaying various aspects of the graphical representation of the dynamics. In addition to these built in palettes, endo will read a user defined palette in the file $HOME/.endomap. In order to specify which color palette is installed, use the "-numwheel #" argument on the command line. By default, endo uses color palette 15 (a rainbow palette). The user defined palette is palette number 0.

The size of the windows can be specified on the command line with the "-W # , # , #" and "-H # , # , #" switches. The first number indicates the width or height of the basin or Lyapunov window. The second (optional) number indicates the width or height of the trajectory-W, -H window. The third (optional) number indicates the width or height of the critical curve -W, -H window.

A "-+" argument causes an overlay plot of the axes. An argument of "-0" indicates to run in demo mode. Arguments of "-1", "-2", "-3" or "-4" indicate display of the y bifurcation diagram, both bifurcation diagrams, 3-D mode, and display of the x bifurcation diagram respectively. An argument of "-6" puts the display into animation mode, erasing the previous image prior to drawing. A color histogram of the trajectory-7 will be displayed if the "-7" switch is given.

An escape radius can be specified with the "-r rval" switch. This radius will only be used if the "Mandelbrot mode" has been selected via "-9 #" where # is 1, 2 or 3. When in Mandelbrot mode, once a trajectory -9 has exceeded the escape radius the iteration is stopped and the point is given a color index from a distinctly different palette.

2.7 Detecting periodic trajectories

In order to turn on periodic orbit detection, use the "-V" argument. This "periodicity" detection is really "near return" detection. To specify the proximity within which an orbit can return before it is classified as periodic, use the "-t " option. If any iterate falls inside a square of size 2 x 2 centered at a previous iterate, it is classified as periodic. The default value of is 0.05.

2.8 Determinant of the Jacobian

Endo uses the determinant of the Jacobian of the map at a point to calculate the critical curves and Lyapunov exponents. Each map in the distributed version of endo has the Jacobian previously calculated in closed form. However, if a new map is added for which the Jacobian is either unknown or difficult to calculate, the "-numeric" option can be used. This option causes endo to numerically approximate the Jacobian. The value of above is used to select points "close" to the point in question.

Another "small" real value, , is used to locate the critical curves. If the absolute value of the determinant of the Jacobian is less than , the point is said to lie on the critical curve. The value of can be specified via the " -z " argument. The default value of is 0.05.

2.9 Saving a window to a file

Each of the five main endo windows may be saved to a file in Portable Pixmap format (PPM). On a monochrome system, the savefile format is Portable Bitmap (PBM). To specify the filename to save to, use the "-o filename" argument. The default filename to use when saving is "endo.out".